The simplest and oldest model for light is that of rays, which can be thought of as particles which move in a straight line and bounce off of reflective surfaces. This model applied to confined regions of space leads to billiard systems, which is described by the rich field of Hamiltonian nonlinear dynamics, developed by Poincare. For reflections of light beams at dielectric surfaces, we can augment the ray model by adding in, in an obvious way, the leading order modification due to wave interference. This modification, the Goos-Haenchen (GH) effect, results in an altered dynamics (still Hamiltonian), which creates a new stable and a new unstable periodic ray trajectory. The stable periodic trajectory corresponds to a standing wave in the optical wave resonator. This "GH mode" was first found in a numerical wave simulation, and its ray explanation given above completes a beautiful demonstration of ray-wave correspondence.